The long-term objective of this project is to provide high-performance crystallographic fast Fourier transform for inclusion in software packages devoted to the visualization of virus or protein structures at atomic resolution. Specifically, the power of new mathematical formulation for crystal symmetric Fourier transformation will be explored as main design tool. The intended result is a highly-efficient mathematical software package for computing three-dimensional crystallographic Fourier transforms on 3D data-samples whose edge-size is a power of a prime (in particular, a power of two) or, product of primes (mixed radix). The routines in this package will avoid computing with redundant data and operations induced by the symmetries of the crystal, and will be applicable to all crystal symmetries. The actual accuracy, applicability, and efficiency of the new methods will be achieved through cycles of computer experimentation and mathematical/algorithmic modifications. Several test problems will be selected, and software prototypes will be tested on different computer platforms.